Fourier Methods for Multidimensional Problems and Backward SDEs in Finance and Economics

In this thesis we deal with processes with uncertainties, such as financial asset prices and the global temperature. Wemodel their evolutions by so-called stochastic processes. Many of these stochastic processes are based on the Wiener process, whose increments are normally distributed. Other models may contain jump components, to model, for example, economic disasters or degradation failures. An important class of models is the Lévy class, where successive increments are independent and statistically identical over different time intervals of the same length. This may give computational advantages.
A well-known application of stochastic processes is in financial mathematics, where the goal is to price financial derivatives or to estimate risk measures. The underlying asset prices may be modeled by, e.g., geometric Brownian motions. More involved models, like the Variance Gamma process, are defined by jumps. In other, for instance economic, personal, or societal, contexts one may face options in the sense of real ‘choices’. For example, should one build a new factory now or in the future? Or should one heighten a dike today, and by how much, or in the future? These decisions are called real options and can often be related to financial options. Similar methods can be used to value them.